The number of colorings of M(n) that are equivalent to their reverse is given by A080107(n). For example, M(4) has A080107(4) = 7 colorings that are equivalent to their reversal: aaaa aabb abab abba abbc abca abcd.

The number of distinct colorings when reversals are counted as equivalent is given by ((A000110(n) + A080107(n))/2, which is essentially the present sequence. M(4) has 11 colorings that are distinct up to reversal: aaaa aaab aaba aabb aabc abab abac abba abbc abca abcd.

We can redo the whole analysis, this time forbidding same-color regions to touch. When we do, we get the same sequences, each with an extra 1 at the beginning. (End)

Note that A056325 gives the number of reversible string structures with n beads using a maximum of six different colors ... and, of course, any limit on the number of colors will be the same as this sequence above up to that number.

If the two ends of the line are distinguishable, so that 'abcb' and 'abac' are distinct, we get the Bell numbers, A000110(n - 1).

With a different offset, number of set partitions of [n] up to reflection (i<->n+1-i). E.g., there are 4 partitions of [3]: 123, 1-23, 13-2, 1-2-3 but not 12-3 because it is the reflection of 1-23. - David Callan, Oct 10 2005